On a simple scheme for computing the electronic energy levels of a finite system from those of the corresponding infinite system.
TL;DR: It is established that this technique, hereafter referred to as the confined Bloch wave (CBW) method, is valid for higher-dimensional symmorphic systems over the entire Brillouin zone, provided some symmetry requirements are satisfied.
Abstract: Computing the electronic energy levels of a finite system or nanostructure is more difficult than computing those of an infinite system or bulk material. In the literature, a technique for simplifying this computation has been proposed, wherein energy levels of a finite system are derived from those of the corresponding infinite system. So far, this method has been validated only for finite length one-dimensional systems and for higher-dimensional systems at k = 0. We establish that this technique, hereafter referred to as the confined Bloch wave (CBW) method, is valid for higher-dimensional symmorphic systems over the entire Brillouin zone, provided some symmetry requirements are satisfied. For this purpose we use a lateral surface superlattice as a model for the infinite system and a stripe or ribbon patterned in this superlattice as a model for the nanostructure. Finally, we compute the subbands of zigzag ribbons of one type patterned in artificial graphene and show that the CBW method predicts all the important subbands in these ribbons, and provides additional insight into the nature of their wavefunctions.
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TL;DR: This work presents a general method to unfold energy bands of supercell calculations to a primitive Brillouin zone using group theoretical techniques, where an isomorphic factor group is introduced to connect the primitive translation group with the supercell translation group via a direct product.
Abstract: We present a general method to unfold energy bands of supercell calculations to a primitive Brillouin zone using group theoretical techniques, where an isomorphic factor group is introduced to connect the primitive translation group with the supercell translation group via a direct product. Originating from the translation group symmetry, our method gives a uniform description of unfolding approaches based on various basis sets and therefore should be easy to implement in both tight-binding models and existing ab initio code packages using different basis sets. This makes the method applicable to a variety of problems involving the use of supercells, such as defects, disorder and interfacial reconstructions. As a realistic example, we calculate electronic properties of a monolayer FeSe on SrTiO in checkerboard and collinear antiferromagnetic spin configurations, illustrating the potential of our method.
42 citations
01 Jan 2017
TL;DR: In this paper, the similarities and differences between the quantum confinements of three-dimensional Bloch waves in one specific direction and the quantum confinement of one-dimensional BLoch waves treated in Chap. 4.
Abstract: In this chapter, we try to understand the similarities and the differences between the quantum confinements of three-dimensional Bloch waves in one specific \(\mathbf{a}_3\) direction and the quantum confinement of one-dimensional Bloch waves treated in Chap. 4. We prove a basic theorem that is the mathematical basis of the theory in this chapter at the beginning, and then we discuss some consequences of this theorem. Afterwards, we obtain the electronic states in several ideal quantum films of different Bravais lattices by reasonings based on this theorem and physical intuition. It is found that in the simplest cases, there are one surface-like subband and \(N_3-1 \) bulk-like subbands in an ideal film of \(N_3\) layers for each bulk energy band. Some understandings obtained are quite different from what are widely believed in the solid-state physics community. A surface state in a multidimensional crystal, in general, does not have to be in a band gap. A thin film of a semiconductor may have a smaller band gap than the bulk semiconductor.
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TL;DR: Gmsh as mentioned in this paper is an open-source 3D finite element grid generator with a build-in CAD engine and post-processor that provides a fast, light and user-friendly meshing tool with parametric input and advanced visualization capabilities.
Abstract: Gmsh is an open-source 3-D finite element grid generator with a build-in CAD engine and post-processor. Its design goal is to provide a fast, light and user-friendly meshing tool with parametric input and advanced visualization capabilities. This paper presents the overall philosophy, the main design choices and some of the original algorithms implemented in Gmsh. Copyright (C) 2009 John Wiley & Sons, Ltd.
5,322 citations
TL;DR: It is found that a non-negligible edge state survives even in graphene ribbons with less developed zigzag edges, when the system size is on a nanometer scale.
Abstract: Finite graphite systems having a zigzag edge exhibit a special edge state. The corresponding energy bands are almost flat at the Fermi level and thereby give a sharp peak in the density of states. The charge density in the edge state is strongly localized on the zigzag edge sites. No such localized state appears in graphite systems having an armchair edge. By utilizing the graphene ribbon model, we discuss the effect of the system size and edge shape on the special edge state. By varying the width of the graphene ribbons, we find that the nanometer size effect is crucial for determining the relative importance of the edge state. We also have extended the graphene ribbon to have edges of a general shape, which is defined as a mixture of zigzag and armchair sites. Examining the relative importance of the edge state for graphene ribbons with general edges, we find that a non-negligible edge state survives even in graphene ribbons with less developed zigzag edges. We demonstrate that such an edge shape with three or four zigzag sites per sequence is sufficient to show an edge state, when the system size is on a nanometer scale. The special characteristics of the edge state play a large role in determining the density of states near the Fermi level for graphite networks on a nanometer scale.
3,834 citations
Book•
13 Dec 2007
TL;DR: In this paper, the authors introduce the concept of representation theory and basic theorems of representation and its application to quantum systems. But they do not discuss the application of double groups to energy bands with spin-orbits.
Abstract: Basic Mathematics.- Basic Mathematical Background: Introduction.- Representation Theory and Basic Theorems.- Character of a Representation.- Basis Functions.- Introductory Application to Quantum Systems.- Splitting of Atomic Orbitals in a Crystal Potential.- Application to Selection Rules and Direct Products.- Molecular Systems.- Electronic States of Molecules and Directed Valence.- Molecular Vibrations, Infrared, and Raman Activity.- Application to Periodic Lattices.- Space Groups in Real Space.- Space Groups in Reciprocal Space and Representations.- Electron and Phonon Dispersion Relation.- Applications to Lattice Vibrations.- Electronic Energy Levels in a Cubic Crystals.- Energy Band Models Based on Symmetry.- Spin-Orbit Interaction in Solids and Double Groups.- Application of Double Groups to Energy Bands with Spin.- Other Symmetries.- Time Reversal Symmetry.- Permutation Groups and Many-Electron States.- Symmetry Properties of Tensors.
366 citations
TL;DR: In this paper, the authors show that modulating a two-dimensional electron gas with a long-wavelength periodic potential with honeycomb symmetry can lead to the creation of isolated massless Dirac points with tunable Fermi velocity.
Abstract: At low energy, electrons in doped graphene sheets behave like massless Dirac fermions with a Fermi velocity, which does not depend on carrier density. Here we show that modulating a two-dimensional electron gas with a long-wavelength periodic potential with honeycomb symmetry can lead to the creation of isolated massless Dirac points with tunable Fermi velocity. We provide detailed theoretical estimates to realize such artificial graphenelike system and discuss an experimental realization in a modulation-doped GaAs quantum well. Ultrahigh-mobility electrons with linearly dispersing bands might open new venues for the studies of Dirac-fermion physics in semiconductors.
165 citations
Book•
07 Nov 2002
TL;DR: In this paper, the authors introduce the concept of femme geometry and describe a set of applications in quantum physics, including quantum mechanical tunneling, quantum waveguides, and boundary element methods.
Abstract: PART 1: INTRODUCTION TO FEM 1. Introduction 2. Simple quantum systems 3. Interpolation polynomials in 1D 4. Adaptive FEM PART 2: 1D APPLICATIONS 5. Quantum mechanical tunneling 6. Schrodinger-Poisson self-consistency 7. Landau states in a magnetic field 8. Wavefunction engineering PART 3: 2D APPLICATIONS OF FEM 9. 2D Elements and shape functions 10. Mesh Generation 11. Applications in atomic physics 12. Quantum wires 13. Quantum waveguides 14. Time dependent problems PART 4: SPARSE MATRIX APPLICATIONS 15. Matrix solvers and related issues PART 5: BOUNDARY ELEMENTS 16. The boundary element method 17. BEM and surface plasmons 18. BEM and quantum applications PART 6: APPENDICES: A. Gaussian quadrature B. Generalized functions C. Green's functions D. Physical constants
163 citations